Solve 1-3/2m/3m-2<0

$$\frac { 1 - \frac { 3 } { 2 } m } { 3 m - 2 } < 0$$

Evaluate (complex solution)

$$m\neq \frac{2}{3}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{1-\frac{3}{2}m}{3m-2}.

$$\frac{\frac{1}{2}\left(-3m+2\right)}{3m-2}<0$$

Extract the negative sign in 2-3m.

$$\frac{-\frac{1}{2}\left(3m-2\right)}{3m-2}<0$$

Cancel out 3m-2 in both numerator and denominator.

$$-\frac{1}{2}<0$$

Compare -\frac{1}{2} and 0.

$$\text{true}$$

Solve for m

$$m\neq \frac{2}{3}$$

Solution Steps

For the quotient to be negative, -\frac{3m}{2}+1 and 3m-2 have to be of the opposite signs. Consider the case when -\frac{3m}{2}+1 is positive and 3m-2 is negative.

$$-\frac{3m}{2}+1>0 3m-2<0$$

The solution satisfying both inequalities is m<\frac{2}{3}.

$$m<\frac{2}{3}$$

Consider the case when 3m-2 is positive and -\frac{3m}{2}+1 is negative.

$$3m-2>0 -\frac{3m}{2}+1<0$$

The solution satisfying both inequalities is m>\frac{2}{3}.

$$m>\frac{2}{3}$$

The final solution is the union of the obtained solutions.

$$m\neq \frac{2}{3}$$